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Properties

Label 2-147-7.3-c6-0-15
Degree $2$
Conductor $147$
Sign $-0.553 + 0.832i$
Analytic cond. $33.8179$
Root an. cond. $5.81532$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.65 − 8.06i)2-s + (−13.5 − 7.79i)3-s + (−11.3 + 19.6i)4-s + (−151. + 87.3i)5-s + 145. i·6-s − 384.·8-s + (121.5 + 210. i)9-s + (1.40e3 + 813. i)10-s + (92.6 − 160. i)11-s + (305. − 176. i)12-s + 3.98e3i·13-s + 2.72e3·15-s + (2.51e3 + 4.35e3i)16-s + (−6.10e3 − 3.52e3i)17-s + (1.13e3 − 1.95e3i)18-s + (4.06e3 − 2.34e3i)19-s + ⋯
L(s)  = 1  + (−0.581 − 1.00i)2-s + (−0.5 − 0.288i)3-s + (−0.176 + 0.306i)4-s + (−1.21 + 0.699i)5-s + 0.671i·6-s − 0.751·8-s + (0.166 + 0.288i)9-s + (1.40 + 0.813i)10-s + (0.0696 − 0.120i)11-s + (0.176 − 0.102i)12-s + 1.81i·13-s + 0.807·15-s + (0.614 + 1.06i)16-s + (−1.24 − 0.716i)17-s + (0.193 − 0.335i)18-s + (0.592 − 0.342i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.553 + 0.832i$
Analytic conductor: \(33.8179\)
Root analytic conductor: \(5.81532\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3),\ -0.553 + 0.832i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.4494608544\)
\(L(\frac12)\) \(\approx\) \(0.4494608544\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.5 + 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (4.65 + 8.06i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (151. - 87.3i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-92.6 + 160. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 3.98e3iT - 4.82e6T^{2} \)
17 \( 1 + (6.10e3 + 3.52e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-4.06e3 + 2.34e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-3.32e3 - 5.75e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 - 1.93e4T + 5.94e8T^{2} \)
31 \( 1 + (2.07e4 + 1.19e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-1.97e4 - 3.42e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 4.62e4iT - 4.75e9T^{2} \)
43 \( 1 + 9.31e4T + 6.32e9T^{2} \)
47 \( 1 + (1.29e5 - 7.47e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-5.09e4 + 8.81e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (1.31e5 + 7.61e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-1.45e5 + 8.42e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (3.05e4 - 5.29e4i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 4.85e5T + 1.28e11T^{2} \)
73 \( 1 + (-7.73e3 - 4.46e3i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (2.21e5 + 3.84e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 5.59e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.79e5 + 1.03e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 1.23e6iT - 8.32e11T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.55335035057219137933991312337, −10.97848319622439368825738434933, −9.681111843288280632857024863595, −8.696895099345852735273891068305, −7.24233154974467916082531689666, −6.46890099139992418212131044704, −4.55959959749671389882157872141, −3.21497055686409498416024634482, −1.84187715147097261076059135143, −0.30972774640889175760339065476, 0.64642817597606383459126205931, 3.33935511872075694349662514298, 4.73317129452206101877038560152, 5.89756037121704034516046815748, 7.14475502181255079616405130101, 8.134307817743838964285429878955, 8.717773287164456385127444741271, 10.12276400660899634774767371011, 11.28500658210830531254351790486, 12.28192541870420048044961821430

Graph of the $Z$-function along the critical line